One path to recovering the information that was lost by thresholding is to imagine that the cell can resolve more details, perhaps distinguishing reliably among three or four different expression levels rather than just two. This is equivalent to imagining that we have multiple readout mechanisms, each of which can only distinguish on/off, but the different mechanisms can have different thresholds, in the spirit of the “French flag” model [1]}. Because we can always put the thresholds in order, having \(K\)  binary switches is the same as distinguishing \(K+1\)  different expression levels. It can be useful to think of thresholding as being implemented at individual binding sites for the TFs, or perhaps at cooperative arrays of binding sites in enhancers, but it is important that our arguments are independent of these microscopic details.
